Harold Hodes

This paper considers two reasons that might support Russell’s choice of a ramified-type theory over a simple-type theory. The first reason is the existence of purported paradoxes that can be formulated in any simple-type language, including an argument that Russell considered in 1903. These arguments depend on certain converse-compositional principles. When we take account of Russell’s doctrine that a propositional function is not a constituent of its values, these principles turn out to be too implausible to make these arguments troubling. The second reason is conditional on a substitutional interpretation of quantification over types other than that of individuals. This reason stands up to investigation: a simple-type language will not sustain such an interpretation, but a ramified-type language will. And there is evidence that Russell was tacitly inclined towards such an interpretation. A strong construal of that interpretation opens a way to make sense of Russell’s simultaneous repudiation of propositions and his willingness to quantify over them. But that way runs into trouble with Russell’s commitment to the finitude of human understanding.

Discourse carries thin commitment to objects of a certain sort iff it says or implies that there are such objects. It carries a thick commitment to such objects iff an account of what determines truth-values for its sentences say or implies that there are such objects. This paper presents two model-theoretic semantics for mathematical discourse, one reflecting thick commitment to mathematical objects, the other reflecting only a thin commitment to them. According to the latter view, for example, the semantic role of number-words is not designation but rather the encoding of cardinality-quantifiers. I also present some reasons for preferring this view.

Logicism is, roughly speaking, the doctrine that mathematics is fancy logic. So getting clear about the nature of logic is a necessary step in an assessment of logicism. Logic is the study of logical concepts, how they are expressed in languages, their semantic values, and the relationships between these things and the rest of our concepts, linguistic expressions, and their semantic values. A logical concept is what can be expressed by a logical constant in a language. So the question “What is logic?” drives us to the question “What is a logical constant?” Though what follows contains some argument, limitations of space constrain me in large part to express my Credo on this topic with the broad brush of bold assertion and some promissory gestures.

Where $\underline{a}$ is a Turing degree and ξ is an ordinal $ , the result of performing ξ jumps on $\underline{a},\underline{a}^{(\xi)}$ , is defined set-theoretically, using Jensen's fine-structure results. This operation appears to be the natural extension through $(\aleph_1)^{L^\underline{a}}$ of the ordinary jump operations. We describe this operation in more degree-theoretic terms, examine how much of it could be defined in degree-theoretic terms and compare it to the single jump operation

Relative to any reasonable frame, satisfiability of modal quantificational formulae in which “= ” is the sole predicate is undecidable; but if we restrict attention to satisfiability in structures with the expanding domain property, satisfiability relative to the familiar frames (K, K4, T, S4, B, S5) is decidable. Furthermore, relative to any reasonable frame, satisfiability for modal quantificational formulae with a single monadic predicate is undecidable ; this improves the result of Kripke concerning formulae with two monadic predicates.

Stewart Shapiro, Foundations without foundationalism: A case for second-order logic. Oxford: Clarendon Press, 1991. xvii + 277 pp. £35.00 A. Diaz, J, Echeverria and A. Ibarra (eds.), Structures in mathematical theories: Reports of the San Sebastian International Symposium, September 25‐29, 1990. Erandio (Vizcaya): Universidad del Pais Vasco, 1990, 492 pp. No price stated J. Echeverria, A. Ibarra, and T, Mormann (eds.), The space of mathematics: philosophical, epistemological, and historical explorations, Berlin and New York: Walter de Gruyter, 1992. xvii + 422 pp., DM 228.00 Jean-Michel Salanskis and Hourya Sinaceur (eds.) Le labyrinthe du continu. Paris: Springer‐Verlag France, 1992. xi + 452 pp. No price stated J. Largeault (ed.) Intuitionisme et théorie de la démonstration. Textes de Bernays, Brouwer, Gentzen, Gödel, Hubert, Kreisel, Weyl. Paris: Vrin, 1992. 556 pp. FF 320 Claude Panaccio, Les mots, les concepts et les choses. La sémantique de Guillaume d’Occam et le nominalisme d’aujourd’hui (Collection Analytiques, volume 3.) Montréal, Bellarmin; Paris, Vrin: 1991. 288 pp. 29,95 Can.$ Desmond Paul Henry, Medieval mereology. (Bochumer Studien zur Philosophie, volume 16.) Amsterdam and Philadelphia: B. R. Grüner, 1991. xxv + 609 pp. Dfl. 240/$130 Stefano Besoli, Il valore della verità, Studio sulla ‘logica della validità’ nelpensiero di Lotze, Firenze: Ponte alle Grazie, 1992, 238 pp. 35000 Lire The Essential Peirce, Selected Philosophical Writings. Volume 1 (1867‐1893). Edited by Nathan Houser and Christian Kloesel. Bloomington and Indian‐apolis: Indiana University Press, 1992. xli + 399 pp. £33.50 (cloth)/£15.99 (paper) Joseph Brent, Charles Sanders Peirce, A Life, Foreword by Thomas Sebeok. Bloomington and Indianapolis: Indiana University Press, 1993. xvi + 386 pp. £28.50 The Collected Papers of Bertrand Russell, Volume 6, Logical and Philosophical Papers 1909‐13, Edited by John G. Slater with the assistance of Bernd Frohmann, London and New York: Routledge, 1992, lxix + 562 pp. £100.00 Francisco Rodriguez‐Consuegra, The mathematical philosophy of Bertrand Russell: Origins and Development. Basel: Birkhäuser Verlag 1991. xiv + 236pp. SFr 98.00 J. Alberto Coffa, The semantic tradition from Kant to Carnap: to the Vienna station, Cambridge: Cambridge University Press, 1991. xi + 445 pp. No price stated Peter Simons. Philosophy and logic in central Europe from Bolzano to Tar ski: selected essays. Dordrecht, Kluwer, 1992. xiv + 441 pp. DA230, $139.00/£79.00 Imre Ruzsa, Intensional logic revisited, Budapest: 1991. viii + 146pp. No price stated W. T. Parry and E. A. Hacker, Aristotelian logic. Albany, NY: State University of New York Press, 1991. x + 545 pp. $49,50 (boards)/$16.95 (paper) P. Bailhache, Essai de logique déontique. Paris: Vrin, 1991. 224 pp. 168F G. Sher, The bounds of logic, A generalized viewpoint, Cambridge, Massachusetts and London, England: The MIT Press, 1991. xv + 178 pp Julian Roberts, The logic of reflection. New Haven and London: Yale University Press, 1992. vii + 307 pp, £25.00 S. Stenlund, Language and Philosophical Problems. London: Routledge, 1990. 231 pp. £35.00 G. Corsi, M, L, Dalla Chiara and G. C, Ghirardi (eds,), Bridging the gap: philosophy, mathematics, and physics, Dordrecht: Kluwer, 1993, xi + 320pp, Dfl. 175/£60.00 Katalin G. Havas, Thought, Language and Reality in Logic. Budapest: Akadémiai Kiadö, 1992. 211 pp. $22.00 K. Devlin, Logic and Information. Cambridge: Cambridge University Press, 1991. xii + 308 pp. £17.95 Bernard Bolzano, Les paradoxes de l’infini, Introduction and translation by Hourya Sinaceur. Paris: Seuil, 1993. 192pp. 140 Ffr Denis Vernant, La philosophie mathématique de Russell. Paris: Vrin, 1993. 509pp. 174 Ffr Anita Burdman Feferman, Politics, logic, and love: The life of Jean van Heijenoort. London: Jones and Bartlett, 1993. xv + 415pp. $29.95.

Where AR is the set of arithmetic Turing degrees, 0 (ω ) is the least member of { $\mathbf{\alpha}^{(2)}|\mathbf{a}$ is an upper bound on AR}. This situation is quite different if we examine HYP, the set of hyperarithmetic degrees. We shall prove (Corollary 1) that there is an a, an upper bound on HYP, whose hyperjump is the degree of Kleene's O. This paper generalizes this example, using an iteration of the jump operation into the transfinite which is based on results of Jensen and is detailed in [3] and [4]. In $\S1$ we review the basic definitions from [3] which are needed to state the general results